An elementary treatise on cubic and quartic curves, by A. B. Basset.

THE FOLIUM OF DESCARTES.

95

cutting CB in B and the circle in A; and on OA take two points.
P and Q such that OP = AB and
1 1 2
+....... (2),
OQ+ OB 'OP........................(
so that OQPB is a harmonic range; then the locus of Q is the:
required curve. If a be the radius of the circle, we have
x2 - y2 = 0Q2 cos 20,
2 + 3y2 = OQ2 (3 - 2 cos2 0),
x-2 y2 cos 20
whence x cos20 (3).
x2 + 3y2 3 - 2 cos2 0.....................
Now OB = a sec 0,
OP= AB = 2a cos0 - a sec 0...............(4),
a cos 20
whence by (2) OQ = cos 0 )
cos 0 (3 - 2 cos2 0)'
and therefore (3) becomes
a(2 - y2) = (x2+ 3y2),
which is the locus of Q.
The locus of P is the logocyclic curve, for if (x, y) are the
coordinates of P,
x2 _ Y2
Y= cos 20,
x2 + y2
and by (4) OP cos 0 = a cos 20.
The form of the curve is almost identical with that of the
logocyclic curve. The origin is a crunode, and the line 3x + a = 0
is the only real asymptote. The curve has one real point of
inflexion which is at infinity, and the asymptote is the inflexional
tangent. To prove this, interchange x and y in (36) of ~ 49, and
it becomes
y (p + qx) (Ix + my + n) + Px3 + Qx2 + Rx + S =0.
In this, put m=P= = -a, Q===-a, = = 0, p = a, q=3
and the equation reduces to (1), and the asymptote 3x + a= 0 is
the inflexional tangent at infinity.